Given a group \(G\) , let \(\operatorname{Cent}(G)\)  denote the set of centralizers of \(G\) , i.e., \(\operatorname{Cent}(G)=\lbrace C(x) \mid x \in G\rbrace \) , where \(C(x)\)  is the centralizer of the element \(x\)  in \(G\) . The study of finite groups in terms of \(|\operatorname{Cent}(G)|\) , becomes an interesting research topic in last few years. It is easy to see that \(|\operatorname{Cent}(G)|=1\)  if and only if \(G\)  is abelian. Belcastro and Sherman proved that there is no group with \(|\operatorname{Cent}(G)|=2\)  or 3 and that for a finite group \(|\operatorname{Cent}(G)|=4\)  if and only if \(\frac{G}{Z(G)} \cong C_2 \times C_2\)  and \(|\operatorname{Cent}(G)|=5\)  if and only if \(\frac{G}{Z(G)} \cong S_3\)  or \(C_3 \times C_3\)  ([1]}). After that the structure of all finite groups are determined for \(|\operatorname{Cent}(G)|=6, 7, 8, 9, 10\)  in terms of \(\frac{G}{Z(G)}\)  (see [2]}, [3]}, [4]}, [5]}, [6]}). There are several other interesting results concerning the impact of \(|\operatorname{Cent}(G)|\) . For example, every finite group with \(|\operatorname{Cent}(G)|\le 21\)  is solvable ([7]}). Again, if \(G\)  is a solvable group (not necessarily finite) of derived length \(d\) , then \(d \le |\operatorname{Cent}(G)|\)  ([8]}). More recently, we have if \(G\)  is a non-trivial group such that \(|\operatorname{Cent}(G)| \ge \frac{2\mid G \mid }{3}\) , then \(G \cong S_3, S_3 \times S_3\)  or \(D_{10}\)  ([9]}) etc. More information on this and related concepts may be found in [10]}, [11]}, [12]}, [13]}, [14]}, [15]}.
